Survey on Affine Spheres
نویسنده
چکیده
Affine spheres were introduced by Ţiţeica in [72, 73], and studied later by Blaschke, Calabi, and Cheng-Yau, among others. These are hypersurfaces in affine R which are related to real Monge-Ampère equations, to projective structures on manifolds, and to the geometry of Calabi-Yau manifolds. In this survey article, we will outline the theory of affine spheres their relationships to these topics. Affine differential geometry is the study of those differential properties of hypersurfaces of R which are invariant under all volumepreserving affine transformations. Affine differential geometry is largely traced to Ţiţeica’s papers in 1908-09, although for curves in R, one of the main invariants, the affine normal, was already introduced by Transon [69] in 1841. Given a smooth hypersurface H ⊂ R, the affine normal ξ is an affine-invariant transverse vector field to H . Define the special affine group as SA(n+ 1,R) = {Φ: x 7→ Ax+ b, detA = 1}.
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